# Pontryagin duality

## 1 Pontryagin dual

Let $G$ be a locally compact abelian group (http://planetmath.org/TopologicalGroup) and $\mathbb{T}$ the 1-torus (http://planetmath.org/NTorus), i.e. the unit circle in $\mathbb{C}$.

Definition - A continuous homomorphism $G\longrightarrow\mathbb{T}$ is called a of $G$. The set of all characters is called the Pontryagin dual of $G$ and is denoted by $\hat{G}$.

Under pointwise multiplication $\hat{G}$ is also an abelian group. Since $\hat{G}$ is a group of functions we can make it a topological group under the compact-open topology (topology of convergence on compact sets).

## 2 Examples

• β’

$\hat{\mathbb{Z}}\cong\mathbb{T}$, via $n\mapsto z^{n}$ with $z\in\mathbb{T}$.

• β’

$\hat{\mathbb{T}}\cong\mathbb{Z}$, via $z\mapsto z^{n}$ with $n\in\mathbb{Z}$.

• β’

$\hat{\mathbb{R}}\cong\mathbb{R}$, via $t\mapsto e^{ist}$ with $s\in\mathbb{R}$.

## 3 Properties

The following are some important of the dual group:

Let $G$ be a locally compact abelian group. We have that

• β’

$\hat{G}$ is also locally compact.

• β’

$\hat{G}$ is second countable if and only if $G$ is second countable.

• β’

$\hat{G}$ is compact if and only if $G$ is discrete.

• β’

$\hat{G}$ is discrete if and only if $G$ is compact.

• β’

$\widehat{(\oplus_{i\in J}G_{i})}\cong\oplus_{i\in J}\hat{G_{i}}$ for any finite set $J$. This isomorphism is natural.

## 4 Pontryagin duality

Let $f:G\longrightarrow H$ be a continuous homomorphism of locally compact abelian groups. We can associate to it a canonical map $\hat{f}:\hat{H}\longrightarrow\hat{G}$ defined by

 $\displaystyle\hat{f}(\phi)\,(s):=\phi(f(s))\;,\qquad\qquad\phi\in\hat{H},\;s\in G$

This canonical construction preserves identity mappings and compositions, i.e. the dualization process $\hat{\;}$ is a functor:

Theorem - The dualization $\hat{\;}:\mathord{\mathbf{LcA}}\longrightarrow\mathord{\mathbf{LcA}}$ is a contravariant functor from the category of locally compact abelian groups to itself.

## 5 Isomorphism with the second dual

Although in general there is not a canonical identification of $G$ with its dual $\hat{G}$, there is a natural isomorphism between $G$ and its dualβs dual $\hat{\hat{G}}$:

Theorem - The map $G\longrightarrow\hat{\hat{G}}$ defined by $s\mapsto\hat{\hat{s}}$, where $\hat{\hat{s}}(\phi):=\phi(s)$, is a natural isomorphism between $G$ and $\hat{\hat{G}}$.

## 6 Applications

The study of dual groups allows one to visualize Fourier series, Fourier transforms and discrete Fourier transforms from a more abstract and unified view-point, providing the for a general definition of Fourier transform. Thus, dual groups and Pontryagin duality are the of the of abstract abelian harmonic analysis.

 Title Pontryagin duality Canonical name PontryaginDuality Date of creation 2013-03-22 17:42:42 Last modified on 2013-03-22 17:42:42 Owner asteroid (17536) Last modified by asteroid (17536) Numerical id 7 Author asteroid (17536) Entry type Theorem Classification msc 43A40 Classification msc 22B05 Classification msc 22D35 Synonym Pontrjagin duality Synonym Pontriagin duality Related topic DualityInMathematics Defines Pontryagin dual Defines Pontrjagin dual Defines Pontriagin dual Defines dual of an abelian group Defines character